On Mems And Henon Type Equations

There are six chapters in this dissertation, which is concerned with a partial differential equation with negative exponent arising from Micro-Electromechanical Systems(MEMS) and a Henon type equations coming from astrophysics.In the first chapter, we will introduce some backgrounds and list some basic knowledge needed in this dissertation, and then state the main results.The second chapter is about a mathematical problem from MEMS model with fringing field, i.e. where λ>0,δ>0are constants, Ω (?) IRN is a smooth bounded domain and g:[0,1)�'IR+satisfies (H): g is a C2, positive, nondecreasing and convex function such that lim g(u)=+∞. The corresponding elliptic problem is that Firstly, we show that the solvability of the stationary problem (7) is characterized by a finite critical value λδ*. That is for any λ∈(0, λδ*), there exists a solution of the problem (7), while if λ> λδ*there is no solution of (7). Meanwhile it is shown that for λ> λδ*, any solution to the parabolic equation will quench (hit the value1) at a finite time Tδ*, which leads to the singularity of the right hand of equation (6). Secondly, we focus on estimating the quenching time Tδ*. We will apply the standard parabolic comparison principle to prove that when λ�'(λδ*)+, the quenching time satisfies Tδ5=O((λ-λδ*)-1/2), provided that the extremal solution of (7) is regular.In Chapter3, we study the following electrostatic MEMS-device parabolic equation with initial data uo(x)∈[0,1): where f is a nonnegative profile, λ>0, p>1. Based on the results in Chapter2, we first analyze a limit behavior of the quenching time T*for the solution u to (8) with the assumption that μ0=0, when λ�'(λ*). We arrive at a result that if Ω is a ball, f(x)=f(|x|), f≤0, then lim T*(λ-λ*)-1/2? is a constant, provided that the extremal solution to the corresponding elliptic equation of (8). Secondly, by means of construction, we show that for any λ<λ*, there exists a suitable function z(x)<1such that if‖u0‖<1,u0(x)≥z(x), and u0(?)z, then the solution u for (8) will quench at a finite time T*. Thirdly, we consider the case where Ω=BR(0), f(x)=1. We give a meaningful result that even though the extremal solution to the corresponding elliptic equation of (8) is singular, there also exits a constant C independent of A such that T*≤C(λ-λ*)-1/2. Finally we are interested in a global existence result of (8). We construct a suitable value M0to ensure that if Ω=BR(0) and f is a constant, for any the solution u to (8) exists globally, provided that U0(x)≤AMo(R2-|x|2).The fourth chapter is concerned with the following biharmonic problem with weight: where f(·):IR�'IR+is a C1, nondecreasing and convex function. We are interested in which conditions of p(x), N and f will lead to the nonexistence or unstability of the solution to (9). Through some integrating by parts we get that when p(x)≡1and N<4, any bounded solution of (9) must be a constant. Next we introduce a useful value and let Then we can verify some conditions of qo and a, under which there doesn’t exist a solution to (9), when N≥4, p(x)=|x|α or (1+|x|)α/2. Furthermore, we show that when N>4, if α and qo satisfy some conditions, then any bounded function u∈C4satisfies Finally we consider the Morse index of equation (9). We will prove that any bounded solution of (4) has infinite Morse index, when N≥5, ρ(x)=|x|α, α>-4,,and whereIn Chapter5, we consider a elliptic systems as follows: where f> Oandg>0are C2nondecreasing and convex function. And ρi(x)=|x|αi, We also define We will show that in some situations,(u, v) doesn’t exist. Moreover, there are some relationships between the existence(or nonexistence) of (u,v) and N, αir,, q0, Q0, where qo:=liminf q(u), Qo:=liminf Q(u). When it comes to the stability of the solution (u, v), it is shown that under some circumstance the solution of (5) is not stable.At last chapter, we will introduce some topics corresponding to this paper and give some questions that we can study and try in the future.